Why do planets have elliptical orbits? And why do some satellites, when launched in lower orbits, go around Earth in elliptical orbits?

At first glance it may seem odd that a force such as gravity, which pulls the planets straight in toward the center of mass, should result in elliptical orbits! But in fact it is quite straightforward to understand why this should be so.

It is certainly possible to set up a satellite so that it has a circular orbit (a circle is just an ellipse whose foci coincide). Gravity can only pull in the direction toward the planet. The inertia of the satellite makes it want to travel in a straight line, but if it does so, its velocity is no longer perfectly perpendicular to the pull of gravity, so gravity pulls it in; this will remove part of the velocity, but as the satellite is also falling inward, it gets a new component of velocity due to the acceleration of gravity. In a circular orbit, we know that the ground speed is constant, so these two effects must perfectly cancel one another out to leave the speed of the satellite unchanged. Now imagine that we fire the satellite's boosters so that its ground speed increases. Now the desire of the satellite to go straight is stronger, so the two effects do not cancel perfectly, and the ground speed will vary. You can see how this corresponds to an elliptical orbit, and how a planet orbiting the Sun behaves in the same way. (Of course, planets have no boosters, but think about what effect the initial velocity of the planet due to the process of its formation would have--what happens if a planet is formed with only a small initial velocity, far from the Sun, or if it is formed with a large velocity, very near to the Sun? What happens if the inital velocity of the planet is zero?).

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